Infinite vs. Finite Space-Bounded Randomized Computations

Probabilistic computations can be very powerful with respect to space complexity, e.g. for logarithmic space, zero probability of error is equivalent to nondeterminism. This power, however, depends on the possibility of infinite computations. A natural open question is if this feature is necessary. We answer the question for sweeping finite automata (SFAs), i.e. two-way finite automata that can change the direction of head motion at endmarkers only. We show that zero probability of error SFAs allowing infinite computations can be exponentially more succinct than zero probability of error SFAs forbidding them. We also provide a strengthened form of this result showing that forbidding infinite computations can not be traded for the more powerful bounded-error probabilistic model. To prove our results, we introduce a technique for proving lower bounds on space complexity of SFAs that generalizes the notion of generic words discovered by M. Sipser.